Perfect Reconstruction Cosine Modulated Filter Banks

By changing the phases $\phi_k$ , the pseudo-QMF filter bank can yield perfect reconstruction:

$$\phi_k = \left(k+\frac{1}{2}\right)\left(L+1\right)\frac{\pi}{2}$$

where $L$ is the length of the polyphase filter ($M=LN$ ).

If $M=2N$ , then this is the oddly-stacked Princen-Bradley filter bank, and the analysis filters are related by cosine modulations of the lowpass prototype:

$$f_k(n) = h(n)\hbox{cos}\left[\left(n+\frac{N+1}{2}\right)\left(k+\frac{1}{2}\right)\frac{\pi}{N}\right],\, k=0:N-1$$

However, the length of the filters $M$ can be any even multiple of $N$ :

$$M=LN, \quad (L/2) \in \cal{Z}$$

L is called the overlapping factor These filter banks are also referred to as Extended Lapped Transforms, when $K \ge 2$ .